30-Jul-1995

Unsolved Problem 31:

Is it always possible to have n points in the plane (no 3 on a line; no 4 on a circle), such that for every k (with 0 < k < n), there is a distance determined by these points that occurs exactly k times?

For example, 4 points determine 6 distances. We want one distance to occur just once, another distance to occur twice, and a third distance to occur three times.

So far, configurations have been found for n=2,3,4,...,8.

Reference:

[Croft 1991]
Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry. Springer-Verlag. New York: 1991. Page 153.
Different Number of Distances
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